Method of using a unidirectional crow gyroscope

ABSTRACT

A method for detecting rotation includes providing a plurality of resonant waveguides generally adjacent to one another and optically coupled to one another. Each resonant waveguide of the plurality of resonant waveguides is configured to allow light to propagate along the resonant waveguide in a planar path. The method further includes propagating light along each path in a clockwise direction or along each path in a counterclockwise direction.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.12/416,767, filed on Apr. 1, 2009 and incorporated in its entirety byreference herein, and which claims the benefit of priority from U.S.Provisional Patent Appl. No. 61/041,567, filed on Apr. 1, 2008, andincorporated in its entirety by reference herein.

BACKGROUND

1. Field of the Invention

This application relates generally to optical waveguide gyroscopes.

2. Description of the Related Art

Recent investigations have shown that slow light can have a profoundimpact on the optical properties of materials and systems. Inparticular, under certain conditions, the sensitivity of interferometricsensors can be in principle greatly enhanced by interrogating theinterferometer with slow light. See, e.g., M. Soljacic, S. G. Johnson,S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystalslow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am.B Vol. 19, 2052-2059 (2002). This intriguing property has a number ofphysical origins, depending on the nature of the waveguide and of theperturbation applied to it. However, irrespective of the exact origin,the sensitivity of a number of different sensors has been shown to scalelike the reciprocal of the group velocity. Since slow light can becharacterized by an extremely large group index (>10⁵), its use can inprinciple improve the sensitivity of optical sensors by many orders ofmagnitude. Although not all sensors benefit in terms of sensitivity fromthis slow-light enhancement, this prospect has far-reaching implicationsfor many applications.

SUMMARY

In certain embodiments, a method of detecting rotation comprisesproviding a plurality of resonant waveguides adjacent to one another andoptically coupled together. Each resonant waveguide of the plurality ofresonant waveguides is configured to allow light to propagate along theresonant waveguide in a planar path. The method further comprisespropagating light along each path in a clockwise direction or along eachpath in a counterclockwise direction.

In certain embodiments, a method of detecting rotation comprisesproviding an optical waveguide gyroscope comprising a plurality ofresonant waveguides adjacent to one another and optically coupled to oneanother. The method further comprises propagating at least a portion ofa first optical signal through the plurality of resonant waveguides suchthat the at least a portion of the first optical signal propagatesthrough each resonant waveguide of the plurality of resonant waveguidesin a clockwise direction. The method further comprises propagating atleast a portion of a second optical signal through the plurality ofresonant waveguides such that the at least a portion of the secondoptical signal propagates through each resonant waveguide of theplurality of resonant waveguides in a counterclockwise direction.

In certain embodiments, a method of detecting rotation comprisesproviding a plurality of resonant waveguides adjacent to one another andoptically coupled together. Each resonant waveguide of the plurality ofresonant waveguides is configured to allow light to propagate along theresonant waveguide in a planar path. The paths of each of the resonantwaveguides of the plurality of waveguides are parallel to one another.At least one of the resonant waveguides includes a twisted portionoptically coupled to an adjacent resonant waveguide. The method furthercomprises propagating light along each path in a clockwise direction oralong each path in a counterclockwise direction.

For purposes of summarizing the invention, certain aspects, advantagesand novel features of the invention have been described herein above. Itis to be understood, however, that not necessarily all such advantagesmay be achieved in accordance with any particular embodiment of theinvention. Thus, the invention may be embodied or carried out in amanner that achieves or optimizes one advantage or group of advantagesas taught herein without necessarily achieving other advantages as maybe taught or suggested herein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A schematically shows a diagram of an example conventional CROWgyroscope configuration with an odd-numbered series (N=3) of low-lossrings.

FIG. 1B schematically shows a diagram of an example conventional CROWgyroscope configuration with a series of seven rings.

FIG. 2 shows a diagram of an example conventional CROW gyroscopeconfiguration with an even-numbered series (N=8) of low-loss rings.

FIG. 3 shows an example conventional CROW gyroscope utilizing a lineararray of coupled rings and separate input and output waveguides.

FIG. 4 shows an example conventional CROW gyroscope utilizing a lineararray of coupled rings and a single input and output waveguide.

FIG. 5A shows an example folded unidirectional Sagnac-based CROWgyroscope with stacked rings in accordance with certain embodimentsdescribed herein.

FIG. 5B shows a conventional fiber optical gyroscope with the same totalarea and loss as the unidirectional Sagnac-based CROW gyroscope withstacked rings of FIG. 5A.

FIG. 6 shows an example unidirectional Sagnac-based CROW gyroscope withtwisted rings in accordance with certain embodiments described herein.

FIG. 7 shows an example unidirectional Sagnac-based CROW gyroscope withsmall interstitial rings in accordance with certain embodimentsdescribed herein.

FIG. 8 shows an example folded unidirectional linear CROW gyroscope withstacked rings in accordance with certain embodiments described herein.

FIG. 9 shows an example unidirectional linear CROW gyroscope withtwisted rings in accordance with certain embodiments described herein.

FIG. 10 shows an example unidirectional linear CROW gyroscope with smallinterstitial rings in accordance with certain embodiments describedherein.

FIG. 11 schematically illustrates an example conventional CROW gyroscopewith three rings rotating at frequency Ω.

FIG. 12 schematically illustrates the matrix formalism for thetransfer-matrix calculations.

FIG. 13 plots a comparison of analytical (approximate) and numerical(exact) results for phase delay.

FIG. 14A schematically illustrates a CROW gyroscope having a pluralityof tightly-packed rings (dots show coupling points between the rings).

FIG. 14B schematically illustrates a FOG having a plurality of stackedrings over one another with an equivalent footprint as the CROWgyroscope of FIG. 12A (dots indicate couplers).

FIG. 15 plots the evolution of output power in the four arms of a CROWgyroscope with N=1 rings.

FIGS. 16A-16C plot the sensitivity of CROW gyroscopes and equivalentFOGs as a function of rotation rate for N=1, 9, 81, respectively.

FIG. 17 plots the response of a maximally flat 3-ring CROW filter.

DETAILED DESCRIPTION

A parameter that is of particular importance to detect with a highaccuracy is rotation rate. Applications of rotation-sensing devices, orgyroscopes, cover a wide range, from inertial navigation of aircraft andautomobiles to stabilization of sea oil platforms, guidance of missilesand rockets, etc. A number of optical gyroscope configurations based oncoupled resonant optical waveguides (CROW) have been proposed andstudied in the literature. See, e.g., J. Scheuer, and A. Yariv, “Sagnaceffect in coupled-resonator slow-light waveguide structures,” Phys. Rev.Lett. Vol. 96, 053901 (2006)); A. Yariv, Y. Xu, R. K. Lee, and A.Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,”Opt. Lett. Vol. 24, 711-713 (1999)); M. S. Shahriar et al., “Ultrahighenhancement in absolute and relative rotation sensing using fast andslow light”, Phys. Rev. A, Vol. 75, 053807 (2007); A. B. Matsko et al.,“Optical gyroscope with whispering gallery mode optical cavities,” Opt.Commun. 233, 107 (2004); Steinberg et al, “Rotation Induced SuperStructure in Slow-Light Waveguides with Mode Degeneracy: OpticalGyroscopes with Exponential Sensitivity,” JOSA B, Vol. 25(5), 1216-1224,May 2007; Chao Peng, Zhengbin Li, and Anshi Xu, “Rotation sensing basedon a slow-light resonating structure with high group dispersion,”Applied Optics Vol. 46, No. 19, 4125-4131, July 2007; and Chao Peng,Zhengbin Li, and Anshi Xu, “Optical gyroscope based on a coupledresonator with the all-optical analogous property of electromagneticallyinduced transparency,” Optics Express Vol. 15, No. 7, 3864-3875, 2 Apr.2007. These references claim that the disclosed gyroscopes have superiorsensitivity, either relative to a single resonator structure (in essencea resonant fiber optic gyroscope (RFOG), see, e.g., H. Lefèvre, TheFiber-Optic Gyroscope, Ch. 11, Artech House, Boston (1993)), or relativeto a conventional fiber optic gyroscope (FOG) (see, e.g., H. Lefèvre,The Fiber-Optic Gyroscope, Ch. 2, Artech House, Boston (1993)). Suchclaims are particularly noteworthy because first the fiber-opticgyroscope (FOG) has been for many years the most successful commercialfiber sensors (see, e.g., G. Pavlath, “Fiber optic gyros: the visionrealized,” in 18th International Optical Fiber Sensors ConferenceTechnical Digest (Optical Society of America, Washington, D.C., 2006),MA3), and second because developing an optical gyroscope with asensitivity greater than possible with a conventional FOG is anappealing prospect that would find several important applications. Thereis consequently a tremendous interest, both scientific and commercial,in developing a gyroscope with much greater sensitivity, and/orlong-term stability, and/or scale factor stability, than the mostsuccessful commercial optical gyroscope to date, which is the FOG.

FIGS. 1A and 1B schematically show two diagrams of conventional CROWgyroscopes of relevance. The CROW gyroscope of FIG. 1A has a loopcomprising a series of N=3 low-loss resonators or resonant waveguides(e.g., rings), and the CROW gyroscope of FIG. 1B has a loop comprising aseries of N=7 resonators or resonant waveguides (e.g., rings). The ringsare optically coupled to each other with a power coupling ratio κ toform an open loop. This loop is closed with a 3-dB loop coupler tocreate a Sagnac interferometer. The loop coupler is optically coupled toa light source and a detector (e.g., via a second optical coupler asshown in FIGS. 1A and 1B). As in a conventional FOG, light is launchedinto the loop coupler, which splits it with equal power into twosignals. One signal travels clockwise (cw) around the Sagnac loop, andthe other one counterclockwise (ccw). For simplicity, the resonantwaveguides can be assumed to be identical rings with the same radius R,and the power coupling coefficient between any two adjacent rings orbetween the rings and the leads can be assumed to be the same, and equalto κ. These conditions are not necessary for the proper operation ofthis Sagnac-based CROW gyroscope, or any of the other CROW gyroscopesdiscussed herein. The conclusions of the following analyses arequalitatively the same when the dimension and/or the couplingcoefficient vary from ring to ring. The exact shape of the loop aroundwhich the N rings are arranged has also no bearing on the overallbehavior of this sensor. For simplicity, it can be assumed throughoutthat the rings are all centered on a circle of radius R₀, and that theleads or injector couplers connecting the rings to the 3-dB couplercomprise circular arcs with the same radius of curvature as the rings(see FIGS. 1A and 1B).

To understand the operating principle of the CROW gyroscope, considerfirst the limit of strong coupling (κ=1). Each ring can support cw andccw modes. For example, the cw mode in ring j couples to the ccw mode inrings j+1 and j−1. The cw light signal then travels about half waythrough the first ring to an optical coupler (e.g., a portion of thefirst ring and the second ring which are strongly optically coupled toone another) at the far end of the first ring, where it is substantiallyfully coupled to the second ring. The same process takes place in eachsubsequent ring, until the signal reaches the 3-dB loop coupler. Thelight signal has therefore traveled around the loop in a “scalloped”pattern that encompasses a certain area B. The ccw signal undergoes thesame process, except that it encounters the rings in the reverse order.So it follows the same optical path, but in the opposite direction.Hence as in a conventional FOG, in the absence of rotation andnonreciprocal effects, the two signals experience the same phase shiftas they travel around the CROW loop in their respective directions.Consequently, when they recombine at the 3-dB coupler, they interfereconstructively into the input arm of the coupler, and all the light(minus what is lost to losses within the loop) is detected at thedetector (see FIGS. 1A and 1B). When the loop is rotated at a rate Ωabout its main axis (i.e., perpendicular to its plane), as a result ofthe Sagnac effect light traveling in the direction of the rotationundergoes a stronger phase shift than light traveling in the oppositedirection. The two signals are no longer in phase, which translates intoa change in the power received by the detector. Except for the scallopedshape of the path followed by the light around the loop, this particularconfiguration behaves exactly the same way as a conventional FOG. Inparticular, it has the exact same sensitivity to rotation as a FOGencircling the same area B.

Note that, in FIGS. 1A and 1B, for the cw and ccw signals to return tothe loop coupler instead of to port C and port D, respectively, which isundesirable, the number of rings is odd. By switching the ports of oneof the two injection couplers, a similar configuration is created thatsupports only an even number of rings, as shown in FIG. 2. Both types ofconfigurations have the same basic properties.

The CROW gyroscope starts differing from a FOG and becomes moreinteresting when κ<1, especially when κ is very weak. Each ring in theSagnac loop is then a high-finesse resonator. The input laser signalfrequency is selected to coincide with a resonance frequency of the(identical) rings at rest (zero rotation). For example, each ring can beconsidered to be a high Q cavity with low coupling to neighboring rings.At each junction or optical coupler between two neighboring rings, avery small fraction κ of the power passes on to the adjacent ring, whilethe rest of the power stays in the same ring. Thus, the effective groupvelocity of a signal propagating from ring to ring can be slow. When thegyroscope is at rest, each signal must thus travel multiple times aroundeach ring, one ring after the other, before reaching the far end of theSagnac loop and interfering with each other. When a rotation is appliedto the Sagnac loop around an axis perpendicular to the plane of theloop, as a result of these multiple passes, each signal accumulates alarger rotation-induced (Sagnac) phase shift in each and every ring thanif it were traveling through each ring only once, and the differentialphase shift is enhanced. The applied rotation therefore results in achange in the optical power coming out of the Sagnac loop at port A (seeFIGS. 1A and 1B), and this power change is larger than if the signalswere not resonating around the rings. This enhancement is expected toscale like the number of times each signal travels around each ring,i.e., like 1/κ. For weak coupling (κ<<1), the increase in sensitivitycan therefore be quite significant, as is the case in a resonant fiberoptic gyroscope (RFOG) (see, e.g., H. Lefèvre, The Fiber-OpticGyroscope, Ch. 11, Artech House, Boston (1993)).

For the CROW gyroscope to work optimally in certain embodiments, (1) allthe rings have a common resonance frequency at all time, (2) thefrequency of the interrogating light remains tuned to this commonresonance frequency at rest, and (3) the linewidth of the light signalis substantially narrower than the linewidth of the resonant modes ofeach resonant waveguide. Achieving these conditions is done using verytight control of the optical path length, index, and transversedimensions of all N rings simultaneously. All of these quantities varywhen portions or all of the rings are subjected to a temperature change,stresses, and/or vibrations. Therefore, controlling these quantitiesconstitutes a significant engineering challenge in practice. However,such challenges can be in principle met with existing technology. Thevery small size of some of the CROW gyroscopes considered forrotation-rate detection (rings of only tens of microns in diameter)helps reduce the temperature variations along the rings, and it reducesthe thermal mass of the interferometer, two conditions that make iteasier to control the temperature of the final structure. Nevertheless,because these structures involve multiple resonant waveguides to bestabilized to the same resonant frequency to an extraordinarily highprecision, this stabilization process remains a serious engineeringdifficulty. Certain embodiments described more fully belowadvantageously satisfy the desired goal of not only offering a highersensitivity to rotation, but also loosening the technical challenges ofstabilizing its optical path.

In the original reference which first disclosed the configuration ofFIGS. 1A and 1B (J. Scheuer and A. Yariv, “Sagnac effect incoupled-resonator slow-light waveguide structures,” Phys. Rev. Lett.Vol. 96, 053901 (2006)), an expression was presented for itsresponsivity or sensitivity to rotation. This sensitivity was found tovary as (N+1)² and as 1/κ. The conclusion that was drawn is that byusing a large number of rings and weak coupling, a CROW gyroscope can bemade considerably more sensitive than a conventional FOG. The term “slowlight” was invoked in that original publication, presumably because theapparent group velocity of the signals traveling through these coupledresonators is greatly reduced (see, e.g., A. Yariv, Y. Xu, R. K. Lee,and A. Scherer, “Coupled-resonator optical waveguide: a proposal andanalysis,” Opt. Lett. Vol. 24, 711-713 (1999)) as the couplingcoefficient κ is varied from 1 to very weak values. In many systems, lowgroup velocity implies high sensitivity, since the phase change can beexpressed as:δφ=L·δk≈L·δω·dk/dω=L·δω/v _(g)where v_(g) is the group velocity. A small group velocity does notimprove the sensitivity to changes in the length L, but it can improvesensitivity to changes in the refractive index n or the frequency ω. Inan optical gyroscope, both signals move with the medium, so δω=0 in thegyroscope frame of reference. Confusion on this point can lead tomistakes (see, e.g., Leonhardt et al., Phys. Rev. A 62, 055801 (2000)).

However, as described more fully below, simulations show that in thisconfiguration, slow light has no net benefit, and that thisconfiguration, when properly scaled for fair comparison to aconventional FOG, exhibits at best the same responsivity or sensitivityas a FOG. These simulations were performed in part to assess the bestpossible rotation sensitivity a CROW gyroscope can have compared to aconventional FOG, assuming) similar gyroscope footprints. Theresponsivity can be expressed as dP(Ω)/dΩ where P(Ω) is power detectedat a rotation rate Ω. These simulations show that although the (N+1)²dependence of the responsivity or sensitivity predicted by Scheuer etal. is correct, it is only applicable when no external phase bias isapplied between the gyroscope's counter-propagating signals. However, itis well known that with such zero phase bias, the sensitivity to firstorder in rotation rate Ω is zero. Hence the power returning to thedetector depends on the next higher (second) order term in the phasedifference between the signals, i.e., on the square of the rotationrate. This configuration has therefore a very poor sensitivity to smallrotation rates, certainly much poorer than that of a conventional FOGhaving a coil of similar size.

One option explored in the performance limitation discussion below is tosupply a suitable phase bias to the CROW gyroscope and hence make thegyroscope signal proportional to Ω. However, as described below, when aproper phase bias is added, the signal does become proportional to Nrather than N². The CROW gyroscope is then more sensitive to smallrotation rates, but its sensitivity is no longer proportional to (N+1)²,but rather to N+1. As a result, the biased CROW gyroscope turns out tohave a sensitivity to rotation comparable to that of a conventional FOGor RFOG with the same footprint and the same total propagation loss whenN=1, and increasingly smaller relative sensitivity as N increases. Thenumber of rings that maximizes the relative sensitivity of the CROWgyroscope of FIGS. 1A and 1B is N=1, and in this configuration the CROWgyroscope looks sensibly like an RFOG. The conclusion is that the CROWgyroscope of FIGS. 1A and 1B has the same one and only benefit over aFOG as a resonant fiber gyroscope, namely it utilizes a shorter lengthof waveguide. However, it also has the same disadvantages over a FOG asan RFOG, namely the aforementioned stringent thermal and mechanicalpath-length stabilization requirements. This significant downside addsgreat engineering complexity and cost, and it certainly does notoutweigh the length advantage. Finally, as discussed below in theperformance limitation discussion below, unlike the situation implied inthe original publication of Scheuer et al., the apparent lower groupvelocity of the light traveling through a CROW gyroscope has no bearingon its responsivity or sensitivity. Unfortunately, the CROW gyroscopebelongs to the class of sensors that are not enhanced by slow light.

This conclusion applies to two other configurations as well, illustratedin FIGS. 3 and 4, discussed by Steinberg et al, “Rotation Induced SuperStructure in Slow-Light Waveguides with Mode Degeneracy: OpticalGyroscopes with Exponential Sensitivity,” JOSA B, Vol. 25(5), 1216-1224,May 2007. In the CROW gyroscope of FIG. 3, the resonators or resonantwaveguides are not arranged in a Sagnac loop but in a straight line.Steinberg et al. also unfortunately makes incorrect predictions aboutthis gyroscope performance. The CROW gyroscope of FIG. 4 is similar; themain difference is that the output signal is collected at the end of thesame waveguide that supplies light to the coupled resonant waveguides.Unlike claimed in Steinberg et al., first the sensitivity of this CROWgyroscope is unrelated to group index. Second, its sensitivity scaleslike the total area of the rings, not like their total length, whichimplies that reducing the rings' radius while keeping their total lengthconstant will yield a reduced sensitivity. Third, the fact that thetransmission varies exponentially with rotation rate has no benefit forthe sensitivity. Fourth, the presence of a bandgap in the transmissionwhen the device is rotated is not a novel effect (it is present inclassical RFOGs), and for equal total loss and area (the proper metricto compare gyroscopes) it does not result in an improvement insensitivity over a conventional FOG. Finally, again for equal total lossand area, a CROW gyroscope has the same sensitivity as an RFOG. Thebottom line is that even if one were to construct a CROW gyroscope withthe world's lowest loss waveguide (silica single-mode fibers around 1.5μm) and hypothetically lossless fiber couplers, the CROW gyroscope ofFIG. 3 would not offer any sensitivity advantage over a conventionalFOG. Such a CROW gyroscope would, on the other hand, present severalsignificant disadvantages, including lack of reciprocity and the need tostabilize the resonance frequencies of N coupled rings againstenvironmental perturbations to an extreme degree (as mentioned inrelation to the CROW gyroscope of FIGS. 1A and 1B).

Several other publications in recent years have described othergyroscope schemes utilizing slow light to purportedly enhance thesensitivity to small rotation rates, and so far most of them have provedto be erroneous. A. B. Matsko et al., “Optical gyroscope with whisperinggallery mode optical cavities,” Opt. Commun. 233, 107 (2004)) introducedanother concept that was also ultimately proved incorrect (see, e.g., A.B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Erratum toOptical Gyroscope with whispering gallery mode optical cavities,” Opt.Commun. 259, 393 (2006); and M. S. Shahriar et al., “Ultrahighenhancement in absolute and relative rotation sensing using fast andslow light”, Phys. Rev. A, Vol. 75, 053807 (2007)).

Certain embodiments described herein provide a number of newunidirectional configurations of CROW gyroscopes in which the lighttravels in the same direction (e.g., cw) in a plurality of adjacentresonant waveguides (e.g., in all or most of the rings). In certainembodiments, these configurations present one significant advantage overall CROW gyroscopes configurations proposed to date, namely a greatersensitivity to rotation rate than a conventional FOG (again, afternormalization to the same area and total loss). In addition, in certainembodiments, some of these configurations are folded, as described ingreater detail below, which means that they exhibit a smaller footprintand hence easier stabilization of their optical path lengths. Theseimprovements offer the new prospect of optical gyroscopes withperformance far superior to that of conventional FOGs. Subsequentpublications (see, Chao Peng, Zhengbin Li, and Anshi Xu, “Rotationsensing based on a slow-light resonating structure with high groupdispersion,”Applied Optics Vol. 46, No. 19, 4125-4131, July 2007; andChao Peng, Zhengbin Li, and Anshi Xu, “Optical gyroscope based on acoupled resonator with the all-optical analogous property ofelectromagnetically induced transparency,” Optics Express Vol. 15, No.7, 3864-3875, 2 Apr. 2007) have also indicated that unidirectionalconfigurations may provide benefits compared to other configurations.

One reason, and perhaps the only reason, why the sensitivity to rotationrate of any of the CROW gyroscopes illustrated in FIGS. 1-4 do notexceed the sensitivity of a conventional FOG is that the light insuccessive rings travels in opposite directions, namely, alternativelycw and ccw. The implication is that the Sagnac phase shift picked up bythe signal as it travels through a given ring is partly cancelled by theSagnac phase shift picked up by the same signal as it travels throughthe adjacent ring. The net Sagnac-induced phase shift accumulated by thelight signal traveling through the structure (e.g., the clockwise signalin the CROW gyroscope of FIGS. 1A and 1B, or the signal traveling upwardthrough the CROW gyroscope of FIG. 4) is then reduced compared to whatit would be if light was to travel in the same direction (e.g., cw), inall or most rings. This can be accomplished by utilizing one of a numberof configurations, shown in FIGS. 5A and 6 through 10.

FIG. 5A schematically illustrates an example optical waveguide gyroscope10 in accordance with certain embodiments described herein. Thegyroscope 10 comprises at least one optical coupler 20 having a firstport 22, a second port 24, and a third port 26. The at least one opticalcoupler 20 is configured to receive a first optical signal 30 at thefirst port 22, to transmit a second optical signal 40 to the second port24, and to transmit a third optical signal 50 to the third port 26. Thegyroscope 10 further comprises a plurality of resonant waveguides 60(e.g., resonant waveguides 60 a, 60 b, 60 c of FIG. 5A) opticallycoupled to the second port 24 and the third port 26. The resonantwaveguides 60 are generally adjacent to one another and are opticallycoupled to one another. At least a portion of the second optical signal40 propagates from the second port 24 to the third port 26 bypropagating through the plurality of resonant waveguides 60. At least aportion of the third optical signal 50 propagates from the third port 26to the second port 24 by propagating through the plurality of resonantwaveguides 60. The at least a portion of the second optical signal 40propagates through each resonant waveguide 60 of the plurality ofresonant waveguides 60 in a clockwise direction and the at least aportion of the third optical signal 50 propagates through each resonantwaveguide 60 of the plurality of resonant waveguides 60 in acounterclockwise direction. In certain embodiments, the at least aportion of the second optical signal 40 is received by the third port 26and the at least a portion of the third optical signal 50 is received bythe second port 24. In certain such embodiments, the at least a portionof the second optical signal 40 and the at least a portion of the thirdoptical signal 50 are combined by the at least one optical coupler 20and transmitted to the first port 22.

In certain embodiments, the gyroscope 10 further comprises at least oneinjector coupler 70 optically coupling the second port 24 to theplurality of resonant waveguides 60 and at least one injector coupler 70optically coupling the third port 26 to the plurality of resonantwaveguides 60. In certain embodiments, the gyroscope 10 furthercomprises a plurality of optical couplers 80, with each optical coupler80 optically coupling two generally adjacent resonant waveguides 60 ofthe plurality of resonant waveguides 60 to one another. In certain suchembodiments, the optical couplers 80 are located between two adjacentresonant waveguides 60 (e.g., between resonant waveguides 60 a and 60 b,and between resonant waveguides 60 b and 60 c).

In certain embodiments, the at least one optical coupler 20 comprises a1×2 optical coupler with three ports or a 2×2 optical coupler with fourports. For example, certain embodiments utilize a 3-dB optical coupler.The at least one optical coupler 20 of certain embodiments splits thefirst optical signal 30 into the second optical signal 40 and the thirdoptical signal 50. For example, in certain embodiments, the at least oneoptical coupler 20 directs 50% of the first optical signal 30 to thesecond optical port 24 as the second optical signal 40 and directs 50%of the first optical signal 30 to the third optical port 26 as the thirdoptical signal 50. In certain embodiments, the at least one opticalcoupler 20 comprises an optical circulator, an optical splitter, orother optical components. In certain other embodiments, free-spacecoupling can be used and the at least one optical coupler 20 comprises apartially transmitting mirror. Other types of optical couplers 20 arealso compatible with certain embodiments described herein.

In certain embodiments, the first optical signal 30 is generated by alight source (not shown). Examples of light sources compatible withcertain embodiments described herein include, but are not limited to,laser sources. In certain embodiments, the light generated by the lightsource has a linewidth significantly smaller than the width of theresonances of the CROW. In certain embodiments in which the at least aportion of the second optical signal 40 and the at least a portion ofthe third optical signal 50 are combined by the at least one opticalcoupler 20, the resultant optical signal is transmitted to an opticaldetector (not shown). Examples of optical detectors compatible withcertain embodiments described herein include, but are not limited to,photodiodes.

In certain embodiments, the resonant waveguides 60 comprise at least onering waveguide which is configured to allow an optical signal toresonate within (e.g., propagate multiple times around) the resonantwaveguide 60. In certain other embodiments, the resonant waveguides 60comprise at least one microresonator. In certain embodiments, theresonant waveguides 60 can be of one or multiple kinds. For example, atleast one of the resonant waveguides 60 can be defined inphotonic-bandgap planar structures, conventional waveguides defined bywell-established methods such as implantation, diffusion,photolithography, etching, etc., in amorphous, crystalline, or ceramicmaterials such as silica, phosphate, fluoride, or other glasses, incrystals such as silicon, or in dielectrics or metals. In certainembodiments, the resonant waveguides 60 can also comprise microspheresor conventional optical fiber (e.g., single-mode optical fiber). Forsilica-based microspheres and silica-based optical fibers, thewavelength range of operation can be about 1.5 μm, a range for which thepropagation loss through the silica-based material is minimal, and hencethe enhancement in sensitivity described above is at a maximum. Examplesof low-loss optical ring resonators compatible with certain embodimentsdescribed herein include, but are not limited to, CaF₂ resonators, whichcan have sufficiently low loss (see, e.g., I. S. Grudinin et al.,“Ultrahigh optical Q factors of crystalline resonators in the linearregime,” Phys. Rev. A 74, 063806 (2006), which is incorporated in itsentirety by reference herein).

Similarly to the conventional CROW gyroscopes of FIGS. 1A and 1B, incertain embodiments, the number N of resonant waveguides 60 can be odd(e.g., 1, 3, 5, 7, 9). Similarly to the conventional CROW gyroscope ofFIG. 2 (e.g., with an appropriate configuration of injector couplers70), in certain other embodiments, the number N of resonant waveguides60 can be even (e.g., 2, 4, 6, 8, 10). In certain embodiments, thecoupling ratio between adjacent resonant waveguides 60 of the pluralityof resonant waveguides 60 is less than one. In certain embodiments, thecoupling ratio is comparable to the round-trip loss of the resonantwaveguide 60. For example, for low-loss ring resonators (e.g.,single-mode fiber at 1.5 μm), the round-trip loss and the coupling ratiocan be less than 10⁻⁴.

In certain embodiments, each resonant waveguide 60 of the plurality ofresonant waveguides 60 is generally planar, and the resonant waveguides60 of the plurality of resonant waveguides 60 are generally parallel toone another. For example, each resonant waveguide 60 of the plurality ofresonant waveguides 60 can define a normal direction generallyperpendicular to the resonant waveguide 60, and the resonant waveguides60 of the plurality of resonant waveguides 60 can be positioned suchthat the normal directions are generally parallel with one another.

In certain embodiments, each resonant waveguide 60 of the plurality ofresonant waveguides is configured to allow light to propagate throughthe resonant waveguide 60 in a generally planar path, and the paths ofeach of the resonant waveguides 60 of the plurality of resonantwaveguides 60 are generally parallel to one another. In certain suchembodiments, each generally planar path has a normal direction generallyperpendicular to the path, and the resonant waveguides 60 are positionedsuch that the normal directions of the paths are generally coincidentwith one another.

In certain embodiments, the resonant waveguides 60 are generally planarwith one another. For example, in certain embodiments, the resonantwaveguides 60 are positioned such that the generally planar paths of theplurality of resonant waveguides 60 are in a common plane. In certainsuch embodiments, the resonant waveguides 60 are stacked above oneanother or folded on top of each other (e.g., the normal directions ofthe resonant waveguides 60 are coincident with one another). Forexample, the gyroscope 10 schematically illustrated by FIG. 5A is afolded unidirectional Sagnac-based CROW gyroscope 10 with stacked rings.In certain embodiments, the resonant waveguides 60 lie substantiallyflat over or on top of each other (e.g., having a zero angle betweenadjacent resonant waveguides 60). In certain other embodiments, at leasttwo adjacent resonant waveguides 60 have a non-zero angle with respectto each other. FIG. 5A shows a non-zero angle between adjacent resonantwaveguides, mostly for clarity of the figure. A small angle betweenresonant waveguides 60, of any value, is acceptable, however the largerthis non-zero angle, the lower the directionality of the gyroscope 10,and the weaker the responsivity or sensitivity of the gyroscope 10.

In certain embodiments (see, e.g., FIG. 5A), the at least one injectorcoupler 70 optically coupling the second port 24 to the plurality ofresonant waveguides 60 comprises a portion of a waveguide sufficientlyclose to a portion of a resonant waveguide 60 to allow at least aportion of an optical signal to propagate between the waveguide portionand the resonant waveguide 60. Similarly, in certain embodiments (see,e.g., FIG. 5A), the at least one injector coupler 70 optically couplingthe third port 26 to the plurality of resonant waveguides 60 comprises aportion of a waveguide sufficiently close to a portion of a resonantwaveguide 60 to allow at least a portion of an optical signal topropagate between the waveguide portion and the resonant waveguide 60.In certain embodiments (see, e.g., FIG. 5A), the plurality of opticalcouplers 80 comprises at least one optical coupler 80 comprisingportions of two adjacent resonant waveguides 60 (e.g., resonantwaveguides 60 a and 60 b) which are sufficiently close to one another toallow at least a portion of an optical signal to propagate between thetwo adjacent resonant waveguides 60.

The principle of the unidirectional CROW gyroscope 10 of FIG. 5A aresimilar to that of the configuration of FIGS. 1A and 1B. To understandthe configuration of FIG. 5A, reference can be made first to the CROWgyroscope of FIGS. 1A and 1B. For simplicity, while it is not arequirement of the CROW gyroscope of FIGS. 1A and 1B, it can be assumedthat the rings or resonant waveguides are all centered on a circle ofradius R₀, as shown in FIGS. 1A and 1B. As demonstrated in theperformance limitation discussion below, the rotation-induced signal inthe gyroscope of FIGS. 1A and 1B originates from two components. One isthe fact that the counter-propagating signals have traveled around alarge loop (circular, and of radius R₀, in this example), and thus theyhave picked up a differential Sagnac phase shift proportional to thearea of this loop, which is labeled B as described above. The secondcontribution is the resonant Sagnac phase shift collected by each of thecw and ccw signals as they propagate through the resonant rings. Theperformance limitation discussion below shows that provided the couplingratio between rings is reasonably low, or equivalently that the finesseof the (identical) ring resonators is sufficiently high, this secondcontribution is much larger than the first contribution.

By folding the rings or resonant waveguides 60 as shown in FIG. 5A, thearea enclosed by the rings (B) has been reduced to zero, so the firstcontribution to the rotation-induced signal vanishes, but since it isnegligible compared to the resonant contribution, the netrotation-induced signal has not been compromised. In fact, the netrotation-induced signal has increased. The reason is that because of theway the resonant waveguides 60 are arranged in this configuration, thesignal traveling, for example, cw around this collapsed loop, travels cwwith respect to the direction of the rotation applied to the structurein every single one of the resonant waveguides 60. For example, asschematically illustrated in FIG. 5A, at least a portion 40 a of thesecond optical signal 40 is coupled from the second port 40 into theresonant waveguide 60 a, at least a portion 40 b is coupled fromresonant waveguide 60 a into the resonant waveguide 60 b, at least aportion 40 c is coupled from resonant waveguide 60 b into resonantwaveguide 60 c, and at least a portion 40 d is coupled from resonantwaveguide 60 c into the third port 26. Each of the portions 40 a, 40 b,and 40 c propagating through the resonant waveguides 60 a, 60 b, and 60c travel is the same direction (e.g., cw) as one another. Thereforeinstead of partially subtracting from each other, all of thesecontributions actually add. In turns, everything else being the same (inparticular the ring size or area, number of rings, waveguide loss, andcoupling ratio 0, this yields a much stronger signal for a givenrotation rate than in the configuration of FIGS. 1A and 1B. Inconclusion, the unidirectional CROW gyroscope 10 of FIG. 5A has agreater sensitivity than the corresponding conventional FOG(schematically illustrated in FIG. 5B) with a length of fiber wrappedaround the same footprint, and with the same total loss.

FIG. 6 schematically illustrates another example optical waveguidegyroscope 10 (e.g., a unidirectional Sagnac-based CROW gyroscope) inaccordance with certain embodiments described herein. The gyroscope 10comprises at least one optical coupler 20 and a plurality of resonantwaveguides 60 (e.g., resonant waveguides 60 a-60 d of FIG. 6) generallyadjacent to one another and optically coupled to one another. In certainembodiments, each resonant waveguide 60 of the plurality of resonantwaveguides 60 is generally planar, and the resonant waveguides 60 of theplurality of resonant waveguides 60 are generally planar with oneanother. At least one optical coupler 80 of the plurality of opticalcouplers 80 comprises a coupling portion 90 of a first resonantwaveguide 60 (e.g., resonant waveguide 60 b) of the plurality ofresonant waveguides 60. The coupling portion 90 is generally adjacentto, and optically coupled to, a second resonant waveguide 60 (e.g.,resonant waveguide 60 a) of the plurality of resonant waveguides 60. Thecoupling portion 90 of certain embodiments comprises two sections of thefirst resonant waveguide 60 that cross over one another. For example, asschematically illustrated in FIG. 6, the coupling portion 90 of resonantwaveguide 60 b comprises a 180 degree twist of the resonant waveguide 60b. In certain such embodiments, the first resonant waveguide 60 isgenerally planar and generally surrounds a first area, and the couplingportion 90 between the twist and the second resonant waveguide 60 isgenerally planar and generally surrounds a second area smaller than thefirst area. For example, in certain embodiments, the second area is lessthan 10% of the first area. In certain embodiments, the second area issmaller than the first area, but not so small as to induce a significantbending loss in the fiber. For example, in a practical fiber-baseddevice, the minimum diameter for the twisted section of the couplingportion 90 is a few millimeters. In certain embodiments, the sizes ofthe twisted sections of the coupling portions 90 are substantially equalto one another so that the resonant waveguides 60 are substantiallyidentical to one another.

FIG. 7 schematically illustrates another example optical waveguidegyroscope 10 (e.g., a unidirectional Sagnac-based CROW gyroscope) inaccordance with certain embodiments described herein. The gyroscope 10comprises at least one optical coupler 20 and a plurality of resonantwaveguides 60 (e.g., resonant waveguides 60 a-60 d of FIG. 6) generallyadjacent to one another and optically coupled to one another. In certainembodiments, each resonant waveguide 60 of the plurality of resonantwaveguides 60 is generally planar, and the resonant waveguides 60 of theplurality of resonant waveguides 60 are generally planar with oneanother. At least one optical coupler 80 of the plurality of opticalcouplers 80 comprises a ring resonator 92 generally adjacent to, andoptically coupled to, a first resonant waveguide 60 (e.g., resonantwaveguide 60 b) of the plurality of resonant waveguides 60 and a secondresonant waveguide 60 (e.g., resonant waveguide 60 a) of the pluralityof resonant waveguides 60. In certain embodiments, the ring resonator 92is generally circular, while in certain other embodiments, the ringresonator 92 has other shapes. In certain embodiments, the firstresonant waveguide 60 is generally planar and generally surrounds afirst area, the second resonant waveguide 60 is generally planar andgenerally surrounds a second area, and the ring resonator 92 isgenerally planar and generally surrounds a third area smaller than thefirst area and smaller than the second area. For example, in certainembodiments, the third area is less than 10% of the first area and thethird area is less than 10% of the second area. In certain embodiments,the third area is smaller than either the first area or the second area,but not so small as to induce a significant bending loss. In certainembodiments, the sizes of the ring resonators 92 are substantially equalto one another.

The above discussion regarding the propagation of optical signalsthrough the gyroscope 10 is applicable to certain embodiments utilizingthe configurations of FIGS. 6 and 7 as well. In FIG. 6, alternatingreversal of the direction of rotation of the optical signals propagatingthrough the ring with respect to the direction of the rotation appliedto the structure is effected by flipping a portion of every other ringby 180° to create a coupling portion 90. In FIG. 7, it is accomplishedby inserting a ring resonator 92 of small diameter between each adjacentring or resonant waveguide 60. The area of the small ring resonator 92is negligible compared to the area of the main rings or resonantwaveguides 60, for example, only 10% of the area of the main rings orresonant waveguides 60.

In certain embodiments, at least one optical coupler 80 of the pluralityof optical couplers 80 comprises the ring resonator 92, a portion of thefirst resonant waveguide 60, and a portion of the second resonantwaveguide 60. In certain such embodiments, the optical coupling betweenthe small ring resonator 92 and one of the portions of the two resonantwaveguides 60 has a low coupling coefficient κ, determined in the samemanner as in the CROW gyroscope of FIGS. 1A and 1B. The optical couplingbetween the small ring resonator 92 and the other portion of the twoportions of the two resonant waveguides 60 has a coupling ratio ofsubstantially 100%. The higher the better; the difference between 100%and the actual value amounts to an undesirable loss, but the device willstill function well. In certain embodiments, this additional loss isequal to or smaller than the value of the round-trip loss of the opticalsignal propagating through a resonant waveguide 60. In this manner,light couples from a first resonant waveguide 60 to the next small ringresonator 92 with a low coupling coefficient κ, and from there on thissignal travels in a strongly-asymmetric “figure-8” through the smallring resonator/second resonant waveguide 60. The process repeats itselfall the way across the structure. As schematically illustrated in FIGS.6 and 7 for the cw-traveling signal (e.g., the second optical signal 40and the portions 40 a, 40 b, 40 c, 40 d propagating around each of theresonant waveguides 60 a, 60 b, 60 c, 60 d, respectively), light inadjacent resonant waveguides 60 clearly travels in the same direction,namely, cw. For the ccw-traveling signal, not shown for clarity, lightin adjacent resonant waveguides 60 also travels in the same direction,but now ccw.

FIG. 8 shows another example optical waveguide gyroscope 10 (e.g., aunidirectional linear CROW gyroscope) with folded or stacked resonantwaveguides 60 or rings in accordance with certain embodiments describedherein. FIG. 9 shows another example optical waveguide gyroscope 10(e.g., a unidirectional linear CROW gyroscope) with twisted couplingportions 90 in accordance with certain embodiments described herein.FIG. 10 shows another example optical waveguide gyroscope 10 (e.g., aunidirectional linear CROW gyroscope) with small interstitial ringresonators 92 in accordance with certain embodiments described herein.In certain embodiments, the plurality of resonant waveguides 60 areoptically coupled to at least one waveguide. For example, asschematically illustrated by FIGS. 8, 9, and 10, one of the resonantwaveguides 60 of the plurality of resonant waveguides 60 is opticallycoupled to a first waveguide 102 and one of the resonant waveguides 60of the plurality of resonant waveguides 60 is optically coupled to asecond waveguide 104. In certain embodiments, the resonant waveguides 60are positioned along a generally straight line, while in certain otherembodiments, the resonant waveguides 60 are positioned along a generallycurved line.

The same discussion regarding the propagation of optical signals throughthe gyroscope 10 applies to certain embodiments utilizing the linearCROW gyroscopes 10 of FIGS. 8, 9, and 10. Here again, the rings orresonant waveguides 60 can be stacked, twisted portions 90, or smallerinterstitial ring resonators 92 can be added as described above to forcelight to circulate in the same direction (e.g., cw) in a plurality ofresonant waveguides 60 (e.g., in all or most of the resonant waveguides60). Similar unidirectional optical waveguide gyroscopes 10 can beconfigured to have a linear array of coupled resonant waveguides 60 witha single waveguide for inputting and outputting signals to the pluralityof resonant waveguides 60, in a manner similar to that of theconventional CROW gyroscope of FIG. 4. Such unidirectional opticalwaveguide gyroscopes 10 look just like the ones shown in FIGS. 8, 9, and10, except that in each case the second waveguide 104 is not present andthe output is read at the port opposite the input port.

In certain embodiments, some of the unidirectional configurationsdescribed above, for example, those of FIGS. 5A and 8 (as well as FIG. 8without the second waveguide 104), also exhibit a significant stabilityimprovement over the other unidirectional configurations. The reason isthat in these gyroscopes, the rings or resonant waveguides 60 arepositioned over (e.g., folded on top of) each other. Consequently, theyphysically occupy a smaller volume, and it is easier to control theirmean temperature and reduce temperature gradients than in othernon-folded configurations. Variations in mean temperature and variationsin temperature gradients are detrimental in certain embodiments of thesesensors because they modify the resonance frequencies of the rings orresonant waveguides 60, which in turn modify the output power at theoutput port of the sensor in a manner that is undistinguishable from avariation in rotation-induced signal.

In certain embodiments, one or more of the CROW gyroscopes 10 shown inFIGS. 5A, 6-10 is operated in a reciprocal mode of operation, namely,through which two signals travel in opposite directions, as well asfolded. As a result of the reciprocal operation, the two signals arenominally exposed to the same external reciprocal perturbations. Suchperturbations include, but are not limited to, refractive index orlength changes in the waveguide(s) due to external temperature changesthat are slow compared to the time of flight of light through thestructure. This effect is well known in FOGs, as described, for examplein H. Lefèvre, The Fiber-Optic Gyroscope, Ch. 6, Artech House, Boston(1993). The structure is then less susceptible to mean temperaturevariations, and to variations in temperature gradients, as well asacoustic waves and vibrations. The folded configuration, as explainedabove, makes it easier to stabilize the structure in temperature. FIG.5A shows an example CROW gyroscope 10 that satisfies this advantageousfeature in accordance with certain embodiments described herein.

Certain other embodiments utilize a CROW gyroscope 10 having aconfiguration that includes the second, but not the first, feature,namely, which are folded (e.g., the CROW gyroscopes 10 of FIGS. 5A and8, and the one of FIG. 8 without the second waveguide 104) but which arenot operated in a reciprocal mode.

Another significant benefit of certain embodiments of the unidirectionalconfigurations described herein is that they can be designed to be muchsmaller than a conventional FOG yet exhibit the same sensitivity. Incertain such embodiments, the smaller size but comparable sensitivity toconventional FOG systems can be accomplished by reducing the number ofcoupled rings or resonant waveguides 60. In other words, althoughcertain embodiments can be used to produce a gyroscope with the samefootprint as a FOG but a superior sensitivity to rotation, it can alsobe designed to produce a gyroscope with either a smaller footprintand/or a smaller number of rings or resonant waveguides but the sameresponsivity or sensitivity. Any configuration between these twoextremes is also possible.

Discussion of Performance Limitations of a Conventional CROW Gyroscope

The following performance limitation discussion (see, M. A. Terrel etal., “Performance Limitation of a Coupled Resonant Optical WaveguideGyroscope,” J. Lightwave Tech., Vol. 27, No. 1, pp. 47-54 (2009), whichis incorporated in its entirety by reference herein), utilizingtheoretical analysis, shows that unlike predicted by others, aconventional unbiased coupled resonant optical waveguide (CROW)gyroscope made of N ring resonators has a responsivity or sensitivity toa rotation rate Ω that is proportional to (NΩ)², and hence itsresponsivity or sensitivity to small rotation rates is vanishinglysmall. In addition, when proper phase bias is applied to theconventional CROW gyroscope, this sensitivity becomes proportional to NΩand is then considerably larger. However, even after optimizing the CROWgyroscope parameters (N and the ring-to-ring coupling coefficient κ),the conventional CROW gyroscope is always less sensitive than aconventional fiber optic gyroscope (FOG) with the same loop loss,detected power, and footprint. This maximum sensitivity is achieved forN=1, i.e., when the conventional CROW gyroscope closely resembles aresonant FOG. The only benefit of a conventional CROW gyroscope istherefore that it utilizes a much shorter length of fiber, by a factorof about 1/(2κ), but at the expense of a stringent control of the rings'optical path lengths, as in a resonant FOG. The slower apparent groupvelocity of light in a conventional CROW gyroscope compared to a FOG isunrelated to this shorter length requirement.

As discussed above, for the conventional CROW gyroscope to workoptimally, (1) all the rings must have a common resonance frequency atall time, (2) the frequency of the interrogating light must remain tunedto this common resonance frequency at rest, and (3) the linewidth of thelight signal must be substantially narrower than the linewidth of theresonant modes of each resonator. Achieving these conditions is doneusing tight control of the optical path length, index, and transversedimensions of all N rings simultaneously, which constitutes asignificant engineering challenge in practice. However, since the goalof this study is to investigate the validity of the claimed superiorityof the ultimate theoretical sensitivity of this sensor, and sincesimilar conditions have been successfully met for the RFOG, in thediscussion below assumes that these conditions are satisfied.

When the device is rotating at frequency Ω (as schematically illustratedby FIG. 11), the signal going clockwise through a ring accumulatesdifferent phase than the signal going counterclockwise through the ring.This is the well-known Sagnac effect. Careful analysis is important whenthe path is not a closed loop. The Sagnac effect is included in thepropagation transfer matrices. In a CROW gyroscope of certainembodiments, a 3-dB coupler divides light into two signals, with onesignal propagates along a path rotating with the device and the othersignal propagates along a path rotating against the device. Aftertraversing the CROW, the signals are recombined, and the Sagnac phaseshift leads to interference between the two signals. Similarly to aregular FOG, a CROW gyroscope is reciprocal in that the two signalspropagate along the same path but in opposite directions (common-moderejection applies). The reciprocal configuration increases stability,and the resonators are kept on resonance by a high stability.

To model the sensitivity to rotation of a CROW gyroscope, thetransfer-matrix method can be used. The transfer-matrix method keepstrack of the total phase each signal accumulates as it propagates fromone coupler to the next inside the CROW gyroscope. For example,referring to FIG. 12, using a matrix formalism, each CROW segment can bebroken into two parts: M_(ring)=M₂M₁. M₁ governs propagation within aring between P₁ and P₂, and M₂ governs transmission from ring j intoring j+1. Additionally, a matrix M_(co) connects the co-propagatinginput and output, and another matrix M_(ctr) connects thecounter-propagating input and output. The overall transfer matrix forthe CROW gyroscope is obtained by multiplying the matrices for eachcomponent of the device. The phase difference between the co-propagatingand counter-propagating signals is a function of the rotation rate ofthe device: Δφ=Δφ(Ω). Note that full matrix analysis shows that theSagnac effect in adjacent rings does not cancel, even though the signalchanges directions from one ring to the next.

The total phase consists of both a rotation-independent component and arotation-dependent component due to the special-relativistic Fresneldrag experienced by a signal in a moving material. (J. Scheuer and A.Yariv, “Sagnac effect in coupled-resonator slow-light waveguidestructures,” Phys. Rev. Lett. Vol. 96, 053901 (2006), and Post, E. J.“Sagnac Effect”, Rev. Mod. Phys. Vol. 39, 475 (1967)) The transfermatrix of each element in the gyroscope (portion of a ring betweencouplers) was expressed as a function of the ring radius R, theeffective index of the ring mode n, the power coupling coefficientbetween rings κ, the rotation rate Ω, and the signal frequency ω. Thematrix of the total gyroscope is then simply the product of thesematrices. For a given input electric field coupled into the gyroscopeloop, this final matrix provides a means for calculating the electricfields interfering at the output coupler. From these two fields, therotation-induced Sagnac phase shift Δφ was then easily determined, foran arbitrarily number of rings N. The sensitivity was then calculated byinserting the Sagnac phase shift in the expression for the basicresponse of a Sagnac interferometer, which takes into account the phasebias of the interferometer.

The product of the transfer matrices was evaluated by one of twomethods. In the first method, MATLAB was used to calculate this productnumerically after assigning a value to each parameter. This approachworks for any arbitrarily high number of rings.

The second method included multiplying the transfer matricessymbolically using Mathematica, which yielded a closed-form analyticalexpression for the output electric field versus rotation rate. Thisapproach allowed the visualization of the analytic dependence of theCROW FOG phase shift on the sensor parameters, in particular κ and N,which provides some guidance into the physics of this FOG. For example,for N=1 and N=3 this approach yielded the following exact expressionsfor the electric field of the co-rotating output signal:

$\begin{matrix}{\mspace{79mu}{{E_{out}^{co}\left( {N = 1} \right)} = \frac{{\kappa\mathbb{e}}^{{\mathbb{i}}\;{{F{({{- {{cn}{({\alpha - {3\pi}})}}} + {3\;{D{({\pi - \alpha})}}R\;\Omega} - {6\; R_{0}\Omega\;{\cos{({\alpha/2})}}}})}}/{({2\pi})}}}}{{\mathbb{e}}^{{\mathbb{i}}\;{FR}\;\Omega} + {\left( {\kappa - 1} \right){\mathbb{e}}^{{\mathbb{i}}\;{Fcn}}}}}} & (1) \\{{E_{out}^{co}\left( {N = 3} \right)} = \frac{\kappa^{2}{\mathbb{e}}^{{\mathbb{i}}\;{{F{({{- {{cn}{({\alpha - {5\pi}})}}} + {5\; R\;{\Omega{({\pi - \alpha})}}} - {10\; R_{0}\Omega\;{\cos{({\alpha/2})}}}})}}/{({2\pi})}}}}{\begin{matrix}{{\mathbb{e}}^{{\mathbb{i}}\; 2{FR}\;\Omega} + {\left( {\kappa - 1} \right)\left( {{2{\mathbb{e}}^{{\mathbb{i}}\;{F{({{cn} + {R\;\Omega}})}}}} - {2\;{\mathbb{e}}^{{\mathbb{i}}\; 2\;{F{({{cn} + {R\;\Omega}})}}}} +} \right.}} \\\left. {{\mathbb{e}}^{{\mathbb{i}}\;{F{({{3\;{cn}} + {R\;\Omega}})}}} + {\mathbb{e}}^{{\mathbb{i}}\;{F{({{cn} + {3\; R\;\Omega}})}}} + {\left( {\kappa - 1} \right){\mathbb{e}}^{{\mathbb{i}}\; 2\;{Fcn}}}} \right)\end{matrix}}} & (2)\end{matrix}$where c is the speed of light in vacuum and F=2πRω/c². The correspondingexpressions for the counter-rotating signal are the same, except that Ωis changed into −Ω. The total electric field at the coupler is the sumof the co- and counter-rotating signals.

This analytical approach becomes increasingly unyielding for largervalues of N, and it was therefore not pursued for values of N largerthan 3. However, in the important limit of small rotation rates(Ω<<Ω₀/(N+1), where Ω₀=κc²ω⁻¹R⁻²), the phase of the electric fieldsgiven in Eqs. (1)-(2) can be expanded in a Taylor series to first orderin ∈=Ω/Ω₀/(N+1)). Comparison between these two expansions, one valid forN=1 and the other for N=3, suggests the following expression for thedependence on N (to first order in ∈) of the Sagnac phase shift:

$\begin{matrix}{{\Delta\phi}_{CROW} = {\frac{4\pi\; R^{2}{\omega\Omega}}{c^{2}}\left( {\frac{N + 1}{2\kappa} + \frac{{2\;{\cot\left( {\alpha/2} \right)}} + \alpha - \pi}{\alpha}} \right)}} & (3)\end{matrix}$where α=2π/(N+2) is the angle subtended by each ring from the center ofthe gyroscope loop. Note that Eq. (3) states that the phase delay isindependent of the refractive index of the fiber. This is in strictagreement with the well-known fact established by Arditty and Lefèvre(H. J. Arditty and H. C. Lefèvre, “Sagnac effect in fiber gyroscopes,”Opt. Lett. Vol. 6, 401-403 (1981)) using relativistic arguments that theresponse of a gyroscope is independent of the material index.

This expression in Eq. (3) is exact for N=1 and N=3. To confirm itsexpected validity for higher values of N (to first order in ∈), FIG. 13plots the approximate phase shift predicted analytically by Eq. 3 versusthe normalized rotation rate Ω/Ω₀ for N=1, 7, 15, 21 and 29 whilekeeping R, ω, and κ constant. For comparison, the exact Sagnac phaseshift is plotted for the same values of N, calculated numerically usingthe first (exact) method. Note that for all practical small rotationrates, the ratio Ω/Ω₀ is extremely small, and the agreement between thetwo models is extremely good. For example, for reasonable CROWparameters (κ=0.001, λ=1.5 μm, and R=1 cm), Ω₀=716 rad/s. For a rotationrate equal to Earth rate (conventional FOGs can routinely detectrotation rates three orders of magnitude smaller than Earth rate),Ω/Ω₀≈10⁻⁷. The maximum ratio Ω/Ω₀ plotted in FIG. 13 (0.003) thereforecorresponds to an extremely large rotation rate (30,000 times Earthrate). FIG. 13 thus shows that even up to fairly high rotation rates,(the agreement improving for smaller rates), the agreement between thetwo models is exceedingly good, even for a large N. This agreement lendscredence to the validity of this useful approximate analytical model.

A number of useful limiting cases can be investigated using Eq. 3. Whenκ=1, the bracket becomes equal to a geometrical form factor which, whenmultiplied by the πR² term in front of the bracket, yields the area B ofthe scalloped region traced by the signals through the coupled rings(see FIGS. 1A and 1B). Equation 3 then becomes:

$\begin{matrix}{{\lim\limits_{\kappa\rightarrow 1}{\Delta\phi}_{CROW}} = \frac{4\; B\;{\omega\Omega}}{c^{2}}} & (4)\end{matrix}$which is exactly the expression of the Sagnac phase shift in a FOG ofarea B. It shows that in this limit of strong coupling, as describedwith physical arguments earlier on, the CROW gyroscope has the samesensitivity as a FOG of same area. This result could not be predictedfrom the expression provided by Scheuer et al. because only the term in1/κ was retained there.

In the opposite limit of weak κ, for any value of N the second term inthe bracket of Eq. 3 becomes negligible compared to the first term. TheSagnac phase shift then becomes:

$\begin{matrix}{{\lim\limits_{\kappa\rightarrow 0}{{\Delta\phi}_{CROW}(N)}} = {\frac{4\pi\; R^{2}\;{\omega\Omega}}{c^{2}}\left( \frac{N + 1}{2\kappa} \right)}} & (5)\end{matrix}$The phase shift is now equal to that of a conventional FOG of area(N+1)πR², but enhanced by factor of 1/(2κ). This term originatesentirely from light resonating around each of the N rings. As expected,it is proportional to area A=πR² of each ring. The effect of theresonant structures is therefore to increase the Sagnac phase shift ininverse proportion to the coupling strength, as in an RFOG. It isinteresting to note that the Sagnac phase shift depends on the number ofrings as (N+1), instead of the expected N. This detail has no impact onthe conclusions of this analysis (if only because of large N thedifference becomes vanishingly small).

To summarize, these two limits and the form of Eq. 3 indicate that inthe CROW gyroscope two contributions are present: a phase shiftindependent of κ that depends on the overall loop area B, as in a FOG,and a resonant phase shift proportional to 1/κ and to the area of eachring A, as in an RFOG. Another important conclusion from Eq. 3 is thatin order to optimize a CROW gyroscope with a given footprint, since theresonant term in the Sagnac phase shift is proportional to the secondpower of R but only the first power of (N+1), the ring radius R, ratherthan the number of rings N, can be increased. Better gyroscopicperformance is obtained when the phase difference Δφ(Ω) is as large aspossible for a given amount of loss. Since Δφ(Ω)∝R² and L_(eff)∝R, alarge radius R is favored. Hence, for a given footprint a CROW gyroscopehas an optimum sensitivity when R is as large as possible, which is whenN=1. Thus, certain CROW gyroscope configurations have the samesensitivity and footprint as do a conventional resonant gyroscope.

The (N+1)²/κ² dependence of the responsivity pointed out by Scheuer etal. arises strictly from the choice of phase bias (zero), a point notspecifically mentioned in by Scheuer et al. In a conventional FOG with azero phase bias, the detected power is proportional to cos²(Δφ2). SinceΔφ is proportional to N_(FOG), the number of turns in the sensing coil,the sensitivity scales like N_(FOG) ². But it also scales like Ω².Consequently, the zero-bias sensitivity exceeds the π/2-bias sensitivityonly for large enough rotation rates, a property that has unfortunatelylittle practical utility. On the other hand, for very slow rotations,which is what most high-accuracy applications utilize,(N_(FOG)Ω)²<<N_(FOG)Ω, and the sensitivity is extremely poor. Thesesimulations show that the same is true for the CROW gyroscope. Toachieve maximum sensitivity, a non-zero bias point can be chosen.

In practice, a CROW gyroscope could be phase biased in much the samemanner as a FOG, namely by placing a phase modulator asymmetrically inthe gyroscope loop, for example at point M in FIGS. 1A and 1B. Thismodulator would then be driven at the proper frequency of the loop,related to the time it takes either signal to go around the loop once (acomplex but quantifiable function of the coupling ratio κ). Studying theeffect of such a dynamic biasing scheme on the sensitivity of the CROWgyroscope would require propagating the two time-dependentcounter-propagating signals through the N coupled rings, which would beextremely time-consuming. As a simpler alternative, the interferometercan be biased by subjecting it to a fixed rotation rate Ω_(b). Althoughthis approach is certainly not practical, it enables the transfer-matrixformalism to be used to quickly yet accurately study the effect of phasebiasing on the sensitivity of a CROW gyroscope. In particular, it canprovide information regarding which bias rotation rate Ω_(b) (and hencewhich phase bias φ_(b)) maximizes the sensitivity to a perturbation δΩin rotation rate. The gyroscope sensitivity is then given by S=dP/dΩ,where P is the power measured by the detector (port A in FIGS. 1A and1B).

Since this analysis is concerned primarily with comparing theperformance of a CROW gyroscope to that of a conventional FOG, it shouldnot unduly put the CROW at either an advantage or disadvantage. Thethree independent design parameters that affect the sensitivity of bothtypes of gyroscopes are (1) the length of waveguide (fiber orotherwise), (2) the waveguide loss per unit length, and (3) the diameterof the sensing loop (which by and large dictates the footprint of thepackaged gyroscope). It would be unfair, for example, to compare a FOGwith a 200-m loop coiled in a 10-cm diameter spool to a CROW gyroscopeutilizing 1000 m of fiber spread on a loop of 10-m diameter, as the CROWwould then have a much greater scale factor, and hence sensitivity.

So for fair comparison, the following analysis applies the followingfour conditions to the two types of gyroscopes. First, the same opticalpower P₀ is assumed to be incident on the two gyroscopes (port A inFIGS. 1A and 1B). Second, the waveguides forming the rings are assumedto have the same low loss as the typical conventional single-mode fiberin a gyroscope (˜0.2 dB/km at 1550 nm). (FOGs typically use apolarization-maintaining fiber, which has a higher loss, but the exactvalue of the loss has no bearing on the result of the comparison.)

Third, their respective effective lengths are imposed to be equal toguarantee that signals suffer the same propagation loss when goingaround the sensing loop. In a conventional FOG, after propagatingthrough N_(FOG) turns of radius R_(FOG) each signal is attenuated by afactor exp(−αL_(FOG)), where L_(FOG)=2πN_(FOG)R_(FOG) is the totalsensing loop length and α is the fiber loss. In a CROW gyroscope, afterpropagating through N rings of radius R, the signal is attenuated by afactor exp(−αL_(CROW)), where L_(CROW) is approximately equal to N timesthe length of one ring (2πR) multiplied by the number of times thesignal goes around each loop (˜1/(2κ)), i.e., L_(CROW)≈NπR/κ). Therequirement for equal gyroscope losses therefore imposesL_(CROW)=L_(FOG), i.e.:

$\begin{matrix}{{N_{FOG}R_{FOG}} = \frac{NR}{2\kappa}} & \left( {6\; a} \right)\end{matrix}$

However, this approximate expression for L_(CROW) is accurate enoughonly for certain ranges of parameters (specifically, when N is large andκ is not too weak). To keep the comparison between the two gyroscopes asaccurate as possible, instead of using this expression, L_(CROW) iscomputed directly by calculating numerically the amount of power P_(A)exiting port A in the non-rotating gyroscope. L_(CROW) is then definedby exp(−αL_(CROW))=P_(A)/P₀, where P₀ is the power incident on the firstloop. Imposing L_(CROW)=L_(FOG) then yielded a condition on N_(FOG) andR_(FOG) slightly different (and more accurate) than Eq. 6(a).

The fourth condition is that two gyroscopes are compared with similarfootprints. Reference to FIGS. 1A and 1B shows that when N is reasonablylarge (more than a few rings), the footprint of a CROW gyroscope inwhich the rings are arranged approximately along a circle is mostlyempty. Since in such a gyroscope the resonant component of the phasesensitivity is independent of the path followed by the rings, one canreduce the footprint of a CROW gyroscope without significantly affectingits sensitivity by arranging the rings along a different path thatbetter utilizes this empty space.

For example, the string of rings can be coiled in a spiral (asschematically illustrated by FIG. 14A) or the rings can be stacked overone another (as schematically illustrated by FIG. 5A). As a result,either more rings can be packed in a circle of given radius, or a givennumber of rings will occupy a smaller footprint. In certain embodiments,a CROW gyroscope can be made shorter than a conventional FOG of the samescale factor, but no shorter than a resonant gyroscope. Note that indoing so, the non-resonant component of the CROW's phase sensitivity(second term in Eq. 3) is greatly reduced. However, for any reasonablylarge value of N this component is negligible compared to the resonantcomponent, so neglecting it does not unduly disfavor the CROW gyroscope.

For simplicity, to calculate the total area of a “coiled” CROW gyroscopeof the type shown in FIG. 14A, this area is assumed to be simply equalto the sum of the area of the N rings, i.e., N_(CROW)pR_(CROW) ². Thisapproximation ignores the area of the interstitial regions between therings, which artificially reduces the CROW gyroscope footprint and henceagain favors the CROW gyroscope over the FOG. Note that thisapproximation is valid not just for a spiral, but for any path thatfills the CROW gyroscope footprint fairly well.

Imposing equal footprint for this “coiled” CROW gyroscope and for theconventional FOG to which it is compared (see FIG. 14B) then yields thefourth condition:N _(CROW) πR _(CROW) ² =πR _(FOG) ²  (6b)

In general, the Sagnac phase shift is proportional to the area aroundwhich light has traveled. For the conventional FOG, this phase shift isdetermined completely by the fiber loss as well as the device footprint.For a CROW gyroscope, the maximum distance that light can travel isstill limited by the fiber loss. Thus, for a same device footprint, themaximal phase shift that a CROW gyroscope can have should be similar tothe corresponding conventional FOG. Intuitively therefore, one shouldnot expect an enhancement of absolute sensitivity in the CROW gyroscopesystem.

Based on the foregoing, the responsivity S versus phase bias of a fewCROW gyroscopes was simulated numerically using the transfer matrixformalism. These gyroscopes all have the same coupling ratio (κ=0.001),ring radius (R=5 cm), and loss coefficient (0.2 dB/km) but differentnumbers of rings N. For each of these CROW gyroscopes, this calculatedsensitivity is compared to that of the “equivalent” FOG, namely the FOGwith a sensing coil of N_(FOG) turns and radius R_(FOG) calculated fromκ, N, and R by imposing L_(CROW)=L_(FOG) and using Eq. 6(b).

In the CROW gyro, as in a simple ring resonator, there exists a criticalcoupling value κ_(crit) which maximizes the power circulating in theindividual rings. However, unlike in a simple ring resonator, thecritical coupling value is not simply given by the single-pass resonatorloss. Our simulations show that (1) the sensitivity of the CROWgyroscope is maximum for a κ value greater than κ_(crit), and (2), novalue of κ makes the CROW gyroscope more sensitive than the equivalentFOG. Hence, for simplicity and without loss of generality, in thefollowing we investigate only the case of coupling greater than criticalcoupling.

The CROW gyroscope suffers from a power-loss mechanism not present in aconventional FOG, namely when its sensing loop is rotated, some of thepower exits the loop at ports C and D (see FIGS. 1A and 1B). This isbecause the individual rings are on resonance when the loop isstationary. Under rotation, the rings are no longer on resonance, andthese two ports transmit some (and equal) amount of power.

To illustrate this point, FIG. 15 plots the power exiting each port,normalized to the input power P₀, as a function of rotation rate for aCROW with the parameters cited above and N=1. At Ω=0, the rings areprobed on resonance and no power exits from ports C and D. Also, as in aclassical FOG, all the power returns into port A, and none in thenonreciprocal port (port B). As Ω is increased, the power in port Adecreases while the power in port B increases, as in a conventionalfiber gyroscope. However, as the cw and ccw signals both slipincreasingly off resonance, some of the power exits at ports C and D.This power leakage increases with Ω, until it is strong enough that thepower in the non-reciprocal port B eventually decreases (see FIG. 15).At large enough rotation rates, the power in port C (and D) reaches aplateau. For some extremely large Ω, well beyond the range of valuescovered by FIG. 15, the next resonance frequency of the loop approachesthe light frequency, and the same process takes place again, in reverse:power drains back out of ports C and D, until at this new resonance allthe power is in port A. FIG. 15 shows that at some rotation rate,identified as Ω_(b), in FIG. 15, the dependence of the power in port Aon Ω, and hence the sensitivity of the CROW gyro, is maximum. Thisconfirms the existence of a phase bias that maximizes the response of aCROW gyroscope.

FIG. 16A shows the sensitivity computed for the same CROW gyroscope withN=1, as well as the sensitivity of the equivalent conventional FOG,again interrogated with the same incident power. As expected from theforegoing discussion, the CROW sensitivity is maximum at Ω_(b) and itdecreases on either side of this optimum value. The phase biascorresponding to this bias rotation rate is φ_(b)≈0.84 rad. This valuedepends weakly on the strength of the coupling between rings. As κ isreduced from the value used in this example (κ=0.001), the resonancesnarrow and hence Ω_(b) decreases. However, light also travels more timesaround each ring, so the actual differential phase shift due to Ω_(b)increases. Simulations shows that these two dependences cancel eachother, as expected, so the optimum phase bias φ_(b) resulting from thisΩ_(b) is essentially independent of coupling strength. It also dependsweakly on the number of rings and on the ring radius. FIG. 16A alsoshows that when the CROW gyroscope is operated at its optimum bias(Ω_(b)≈=2.1 rad/s) its sensitivity is essentially the same (within 3%)as that of the equivalent FOG. The small difference in sensitivity isdue to the different shapes of the transmission functions for the CROWand FOG, and it cannot be significantly enhanced by changing the CROWparameters.

FIGS. 16B and 16C show the same curves for N=9 and N=81, respectively.As N increases, the maximum sensitivity of the equivalent FOG gyroscopeincreases. The reason is that the product R_(FOG)N_(FOG) increases withN (see Eq. 6a), hence the scale factor of the equivalent FOG, whichrelates the Sagnac phase shift (or S) to Ω and which is proportional toR_(FOG) ²N_(FOG), increases. The sensitivity of the CROW gyroscope alsoincreases with N, but it does so more slowly than the FOG, so thesensitivity of the CROW gyroscope relative to the FOG decreases. This isconsistent with the prediction that the sensitivity of the CROWgyroscope compares most favorably to that of the equivalent FOG for N=1.Simulations show that this ratio of sensitivities is approximatelyconstant for κ between ˜10⁻⁴ and ˜10⁻¹. For κ<10⁻⁴ or κ>0.1, this ratiodecreases. As stated above, the ratio of maximum sensitivities cannot besignificantly enhanced by changing κ.

It is interesting to note that the dependence of the ratio of FOG toCROW sensitivity can be predicted from basic principles, using Eq. 3. Ingeneral, the maximum sensitivity of a gyroscope is proportional to theeffective area it covers. For a FOG, this is simply N_(FOG)πR² _(FOG).From Eq. 3, the effective area covered by the CROW rings isapproximately N_(CROW)πR² _(CROW)/(2κ) for large values of N. The firstterm in Eq. 3 can be ignored since it is the nonresonant term thatdepends on the overall path traced by the CROW. Using these definitionsof effective area, as well as Eq. 6, it is trivial to show that for thesame loss, the equivalent FOG has an effective area (and hence a maximumsensitivity) N^(1/2) _(CROW) times greater than the CROW. FIG. 16A-16Cshows that this approximate expression works quite well. Intuitively,the FOG is more sensitive than the equivalent CROW because a signaltraveling along a large loop (as in the FOG) accumulates morerotation-induced phase per unit length than a signal traveling along asmall loop (as in the CROW).

These conclusions were drawn for a particular example, but they areindependent of the choice of parameter values. The overall conclusion isthat the CROW gyroscope offers no significant enhancement insmall-rotation sensitivity compared to a conventional FOG.

The above analysis suggests that although the light travels through aCROW gyroscope with an apparent group velocity that is lower than in anon-resonant waveguide, slow light plays no role in the enhancedresponse of a CROW gyroscope. To illustrate this important point,consider the behavior of the CROW gyroscope shown in FIGS. 1A and 1Bwhen the radius of each ring approaches zero while keeping the overallarea B covered by the loop of rings constant. In this case, the numberof rings increases indefinitely, the total area covered by theindividual ring resonators goes to zero, and the sensing loop convergesto a circle of constant area B. It is then easy to show from Eq. 3 (andconfirm with exact simulations) that the differential phase shiftapproaches 4BωΩ/c², which is precisely the differential phase shift of aFOG of area B. Yet in this CROW, the apparent group velocity of thelight is much slower than in this FOG, in which light travels with a“normal” group velocity. Specifically, it is easy to show that thisgroup velocity is roughly cκ/(πn), independently of N. The twogyroscopes have very different apparent group velocity yet the samephase sensitivity to rotation.

In certain embodiments, the CROW configuration can be used as a filter.FIG. 17 plots the response of a maximally flat 3-ring CROW filter. Bysuitably choosing the coupling coefficients, the CROW structure ofcertain embodiments can be made into flat filters with steep sides.

Various embodiments of the present invention have been described above.Although this invention has been described with reference to thesespecific embodiments, the descriptions are intended to be illustrativeof the invention and are not intended to be limiting. Variousmodifications and applications may occur to those skilled in the artwithout departing from the true spirit and scope of the invention asdefined in the appended claims.

What is claimed is:
 1. A method of detecting rotation, the methodcomprising: providing a plurality of resonant waveguides adjacent to oneanother and optically coupled together, wherein each resonant waveguideof the plurality of resonant waveguides is configured to allow light topropagate along the resonant waveguide in a planar path that surroundsan area substantially bounded by the resonant waveguide; and propagatingat least a portion of a light signal sequentially along each path in aclockwise direction relative to the area surrounded by the path or alongeach path in a counterclockwise direction relative to the areasurrounded by the path.
 2. The method of claim 1, wherein the pluralityof resonant waveguides comprises a first resonant waveguide configuredto allow light to propagate along the first resonant waveguide in afirst path surrounding a first area and a second resonant waveguideadjacent to the first resonant waveguide and configured to allow lightto propagate along the second resonant waveguide in a second pathsurrounding a second area, and said propagating at least the portion ofthe light signal comprises propagating the light either successivelythrough the first resonant waveguide along the first path in a clockwisedirection relative to the first area and through the second resonantwaveguide along the second path in a clockwise direction relative to thesecond area, or successively through the first resonant waveguide alongthe first path in a counterclockwise direction relative to the firstarea and through the second resonant waveguide along the second path ina counterclockwise direction relative to the second area.
 3. The methodof claim 1, wherein at least one of the resonant waveguides comprises atleast one ring waveguide.
 4. The method of claim 1, wherein at least oneof the resonant waveguides comprises at least one microresonator.
 5. Themethod of claim 1, wherein the resonant waveguides are positioned suchthat the planar paths of the plurality of resonant waveguides are in acommon plane.
 6. The method of claim 5, wherein the resonant waveguidesare positioned along a straight line.
 7. The method of claim 5, whereinthe resonant waveguides are positioned along a curved line.
 8. Themethod of claim 1, wherein each planar path has a normal directionperpendicular to the path, and the resonant waveguides are positionedsuch that the normal directions of the paths are coincident with oneanother.
 9. The method of claim 1, wherein at least one of the resonantwaveguides includes a twisted portion optically coupled to an adjacentresonant waveguide.
 10. The method of claim 1, wherein at least one ofthe resonant waveguides is optically coupled to an adjacent resonantwaveguide by a ring waveguide having an area smaller than the area ofthe at least one of the resonant waveguides.
 11. The method of claim 1,wherein a coupling ratio between adjacent resonant waveguides of theplurality of resonant waveguides is less than one.
 12. The method ofclaim 1, wherein adjacent resonant waveguides of the plurality ofresonant waveguides are optically coupled by an optical coupler disposedbetween the adjacent resonant waveguides.
 13. The method of claim 12,wherein the optical coupler comprises a ring resonator, a twistedportion of a resonant waveguide of the adjacent resonant waveguides, orportions of the adjacent resonant waveguides spaced sufficiently closeto one another to allow light to propagate between the adjacent resonantwaveguides.
 14. A method of detecting rotation, the method comprising:providing an optical waveguide gyroscope comprising a plurality ofresonant waveguides adjacent to one another and optically coupled to oneanother; propagating at least a portion of a first optical signalthrough the plurality of resonant waveguides such that the at least aportion of the first optical signal propagates sequentially through eachresonant waveguide of the plurality of resonant waveguides in aclockwise direction relative to an area surrounded and substantiallybounded by the resonant waveguide; propagating at least a portion of asecond optical signal through the plurality of resonant waveguides suchthat the at least a portion of the second optical signal propagatessequentially through each resonant waveguide of the plurality ofresonant waveguides in a counterclockwise direction relative to the areasurrounded and substantially bounded by the resonant waveguide.
 15. Themethod of claim 14, wherein a first resonant waveguide and a secondresonant waveguide of the plurality of resonant waveguides are adjacentto one another and are optically coupled to one another by a couplingportion of the first resonant waveguide, wherein the coupling portioncomprising two sections of the first resonant waveguide that cross overone another.
 16. The method of claim 15, wherein the coupling portioncomprises a 180 degree twist of the first resonant waveguide, the firstresonant waveguide is planar and surrounds a first area, the couplingportion between the twist and the second resonant waveguide is planarand surrounds a second area less than 10% of the first area.
 17. Themethod of claim 14, wherein a first resonant waveguide and a secondresonant waveguide of the plurality of resonant waveguides are adjacentto one another, the first resonant waveguide and the second resonantwaveguide are optically coupled to one another by a ring resonator. 18.The method of claim 17, wherein the first resonant waveguide is planarand surrounds a first area, the second resonant waveguide is planar andsurrounds a second area, and the ring resonator is generally planar andgenerally surrounds a third area, the third area smaller than the firstarea and smaller than the second area.
 19. The method of claim 18,wherein the third area is less than 10% of the first area and less than10% of the second area.
 20. The method of claim 14, further comprisingcombining the at least a portion of the first optical signal afterpropagating through the plurality of resonant waveguides and the atleast a portion of the second optical signal after propagating throughthe plurality of resonant waveguides.
 21. The method of claim 14,wherein each resonant waveguide of the plurality of resonant waveguidesis planar, and the resonant waveguides are parallel to one another. 22.The method of claim 21, wherein the resonant waveguides are planar withone another.
 23. The method of claim 22, wherein the resonant waveguidesare positioned along a straight line.
 24. The method of claim 22,wherein the resonant waveguides are positioned along a curved line. 25.The method of claim 21, wherein each resonant waveguide defines a normaldirection perpendicular to the resonant waveguide, and the resonantwaveguides are positioned such that the normal directions are coincidentwith one another.
 26. The method of claim 21, wherein the resonantwaveguides are stacked above one another.